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Singularities of Steady Axisymmetric Free Surface Flows with Gravity
Author(s) -
Varvaruca Eugen,
S. Weiss Georg
Publication year - 2014
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21514
Subject(s) - inviscid flow , mathematics , scaling , euler equations , conservative vector field , stagnation point , mathematical analysis , gravitational singularity , axial symmetry , vector field , rotational symmetry , invariant (physics) , compressibility , classical mechanics , geometry , physics , mathematical physics , mechanics , heat transfer
We consider a steady axisymmetric solution of the Euler equations describing the irrotational flow without swirl of an incompressible inviscid fluid acted on by gravity and with a free surface. We analyze stagnation points as well as points on the axis of symmetry. At points on the axis of symmetry that are not stagnation points, constant velocity motion is the only blowup profile consistent with the invariant scaling of the equation. This suggests the presence of downward‐pointing cusps at those points. At stagnation points on the axis of symmetry, the unique blowup profile consistent with the invariant scaling of the equation is the Garabedian pointed bubble solution with water above air. Thus at stagnation points on the axis of symmetry with no water above the stagnation point, the invariant scaling of the equation cannot be the right scaling. A finer blowup analysis of the velocity field yields that in the case when the surface is described by an injective curve, the velocity field scales almost likeX 2 + Y 2 + Z 2and is asymptotically given by V ( X , Y , Z ) = c 0 ( X , Y , − 2 Z ) , with a nonzero constant c 0 . The last result relies on a frequency formula in combination with a concentration compactness result for the axially symmetric Euler equations by Delort. While the concentration compactness result alone does not lead to strong convergence in general, we prove the convergence to be strong in our application. © 2014 Wiley Periodicals, Inc.